# FRAMING A CUBIC EQUATION WITH GIVEN THE ROOTS

## About "Framing a Cubic Equation with Given the Roots"

Framing a Cubic Equation with Given the Roots :

Here we are going to see how to construct a cubic equation with given roots.

Let us consider a general cubic equation

ax3 + bx2 + cx + d = 0

α + β + γ  =  -b/a

α β + β γ + γα  =  c/a

α β γ  =  -d/a

Since the degree of the polynomial equation is 3, we have  0 and hence division by a is meaningful. If a monic cubic polynomial has roots α , β , and γ , then

coefficient of x2  =  − (α + β + γ)

coefficient of x  =  α β + β γ + γα

constant term  =  −α β γ

## Framing a Cubic Equation with Given the Roots - Practice questions

Question 1 :

If α , β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0 , form a cubic equation whose roots are

(i) 2α , 2β , 2γ

Solution :

x3 + 2x2 + 3x + 4 = 0

By comparing the  given equation with the general form of cubic equation, we get

a  =  1, b  =  2, c  =  3 and d  =  4.

Coefficient of x2  =  α + β + γ   =  2

Coefficient of x  =   α β + β γ + γα   =  3

Constant term  =  α β γ   =  4

Here α = 2α, β = 2β, γ =

Coefficient of x2  =  2α + 2β + 2γ

2(α + β + γ)  =  2(2)  =  4

Coefficient of x  =  2α (2β) + 2β(2γ) + 2γ(2α)

=  4 (αβ) + 4(βγ) + 4(γα)

=  4 (αβ + βγ + γα)

=  4(3)

=  12

Constant term  =  2α (2β) (2γ)

=  8 α β γ

=  8(4)

=  32

Hence the required equation is

x3 + 4x2 + 12x + 32 = 0

(ii)  1/α, 1/β, 1/γ

Coefficient of x2  =  (1/α) + (1/β) + (1/γ)

(β γ + α γ + α β)/α β γ  =  3/4

Coefficient of x  =  (1/α) (1/β) + (1/β) (1/γ) + (1/γ)(1/α)

=  (1/α β) + (1/βγ) + (1/γα)

=  (γ + α + β)/α β γ

=  2/4  =  1/2

Constant term  =   (1/α) (1/β) (1/γ)

=  1/αβγ

=  1/4

By applying the above values in the general form of the cubic equation, we get

x3 + (3/4)x2 + (1/2)x + (1/4) = 0

4x3 + 3x2 + 2x + 1 = 0

Hence the required cubic equation is 4x3 + 3x2 + 2x + 1 = 0.

(iii) −α, -β, -γ

Coefficient of x2  =  −α + (-β) + (-γ)

=  - (α + β + γ)

=  -2

Coefficient of x  =  −α (-β) + (-β)(-γ) + (-γ) (−α)

=  α β + β γ + α γ

=  3

Constant term  =  −α (-β) (-γ)

=  - αβγ

=  -4

By applying the above values in the general form of the cubic equation, we get

x3 + (-2)x2 + 3x + (-4) = 0

x3 - 2x2 + 3x - 4 = 0

Hence the required cubic equation is x3 - 2x2 + 3x - 4 = 0.

After having gone through the stuff given above, we hope that the students would have understood, "Framing a Cubic Equation with Given the Roots".

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